aircraft mission lecture 8 aircraft...

Aircraft Mission

Constraints analysisIntroduction

Concept of Constraints

Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases

Mission analysisIntroduction

Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA

Takeoff weight estimation

Conclusions

8.108

Lecture 8Aircraft MissionText:

Motori AeronauticiMar. 22, 2016

Mauro ValoraniUniversità La Sapienza

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.109

Agenda

1 Constraints analysisIntroductionConcept of ConstraintsMathematical model

Aerodynamic PolarThrottle Lapse

Flight phases

2 Mission analysisIntroductionAircraft weightsCruise weight ratio

TSFC behavior

BCM/BCATakeoff weight estimation

3 Conclusions

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.110

The difficulties of engine design

Gas Turbine engines exert a dominant influence on aircraft performanceand must be custom tailored for each specific application.

⇒ Engine Specifications come from Aircraft Specifications

The design process is both started by and constrained by an identifiedneed

The process is inherently iterative

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.111

The need : Request for Proposal (RFP)

It’s the mission specification that defines the desired engineperformance.

The aircraft customer describes the desired aircraft performance in adocument such as a Request for Proposal

Example:1 Takeoff, field is at 2000 ft pressure altitude. Takeoff ground roll must be

less than 2500 m at MTOW2 Takeoff rate of climb greater than 1000ft/min3 Subsonic cruise at Best Cruise Mach, maximum range 10000 km4 Payload of 60000 kg

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.112

Design Process

Figure: Design process, schematic

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.113

A Roadmap

DESIGN PROCESS⇒ Constraint and Mission AnalysisChoice of (TSL/WTO) and (WTO/S)Estimation of WTO to obtain TSL

ENGINE SELECTION⇒ Parametric Cycle Analysis and Performance

ENGINE COMPONENTS⇒ Components Sizing

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.114

Design Process

MASS FLOW

CONSTRAINT&

MISSIONANALYSIS

PARAMETRICCYCLE

ANALYSIS

Mission Specs Efficiencies (1st attempt)

Thrust Cycle parameters (βc, T4, BPR, …)

Specific Thrust Ia

Componentsizing

Assumed TSFCbehavior with

h, V, δT

OFF-DESIGN

Geometries

Efficiencies (Actual)

Cross-section,blade profiles,combustor, …

Desired TSFCReference flight conditionTech limitation

Figure: Design process, schematic

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.115

The concept of constraints

The requirements of the RFP can be converted into a series of functionalrelationships between:

the thrust-to-weight ratio at sea-level takeoff

TSL/WTO

the wing loading at takeoffWTO/S

We are looking for equations of the kind:

TSL/WTO = f (WTO/S)

for each of the requirements (flight phases).

These will represent constraints that have to be attained simultaneously.

Of course, many legitimate solutions exist, and none can be identified asoptimal or unique.

The "best" solution is always given by judgment and compromise.

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.116

Constraints Diagram

Each requirement gives life to a curve in the constraint diagram.

The solution space is the region above all the curves

For a given WTO , a low WTO/S means large wing area and increased drag,while a high TSL/WTO results in a large thrust requirement. One may prefer,therefore, relatively low thrust and high wing loadings.

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.117

Constraints DiagramDesign points of actual passenger/cargo aircrafts.

The selected design point is very sensitive to the application and thepreferences of the designer.

1 lbfft2

= 47.88 Nm2

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.118

Constraints Diagram

Design points of actual fighter aircrafts.

1 lbfft2

= 47.88 Nm2

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.119

Master equation

The design process starts by considering the forces that act on the aircraft(modeled as a point mass): lift, drag, thrust, weight.

Equation of motion in the velocity direction:

T cos(AOA + ϕ)− D −W sin(θ) =Wg0

dVdt

(15)

where AOA is the angle between Velocity and Wing Chord Line, ϕ is the anglebetween Wing Chord Line and Thrust axis.Multiplying by the velocity V, we obtain the energy conservation equation:

(T cos(AOA + ϕ)− D)V = W

(V sin(θ) +

ddt

(V 2

2g0

))(16)

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.120

Master equation

Assuming small angles of attack (AOA ≈ 0) and small thrust vectormisalignments with V (ϕ ≈ 0), and recalling that V Sinθ = dh

dt :

V(T − D)

W=

d(

V 2

2g0+ h)

dt=

dze

dt= Ps (17)

where ze represents the aircraft mechanical energy (kinetic + potential) andis often referred to as "energy height".

Ps is the time rate of change of the energy height and is called weight specificexcess power.

Isolating the thrust-to-weight ratio at LHS:

TW

=DW

+Ps

V(18)

Here, both T and W depend on the flight condition and mission phase.

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.121

Master equation: assumptions for T and W

It is assumed that the installed thrust and the actual aircraft weight are given by(SL=Sea Level Static, TO=Take-Off):

T = αTSL (19)

W = βWTO (20)

where α is the full throttle thrust lapse (dependent on altitude, speed andafterburner on/off) and β depends on how much fuel has been consumed.

The equation becomes:

TSL

WTO=β

α

(D

βWTO+

Ps

V

)(21)

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.122

Aerodynamic polar

Recall that lift can be expressed through the lift coefficient as follows:

L = nW =12ρV 2SCL = qSCL q :=

12ρV 2 (22)

⇒ CL =nWqS

=nβq

WTO

S(23)

and that also drag has a similar expression:

D = qCDS (24)

where CD can be expressed through the aerodynamic lift-drag polar:

CD = CD0 + K1C2L + K2CL (25)

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.123

Aerodynamic polar

Conventional form:

CD = CDmin + K ′C2L + K ′′(CL − CLmin )2 (26)

where:

K ′ is the induced drag (inviscid drag due to lift)

K ′′ is the skin+pressure drag (viscous drag due to lift)

Expanding:

CD = (K ′ + K ′′)C2L − (2K ′′CLmin )CL + (CDmin + K ′′C2

Lmin)

or:CD = CD0 + K1C2

L + K2CL (27)

where

K1 = K ′ + K ′′

K2 = −2K ′′CLmin

CD0 = CDmin + K ′′C2Lmin

Assumptions widely used:

K1 = 1πARe

K2 ' 0

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.124

The equation:TSL

WTO=β

α

(D

βWTO+

Ps

V

)(28)

becomes, using the expression D = qCDS and the aerodynamic polar:

TSL

WTO=β

α

{q

β(WTO/S)

[k1C2

L + k2CL + CD0

]+

Ps

V

}(29)

and using the expression for CL = nβq

WTOS :

TSLWTO

= βα

{q

β(WTO/S)

[k1

(nβq (WTO/S)

)2+ k2

(nβq (WTO/S)

)+ CD0

]+ Ps

V

}(30)

This equation is the sought after TSL/WTO = f (WTO/S) which depends on

flight conditions (α, β, V , ρ, n, PS)

aircraft aerodynamic features (CD0 , k1[AR, e], k2)

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.125

Throttle Lapse (α)

The available thrust is expressed as:

T = αTSL

The meaning of α is NOT that of a throttle setting, but rather that of aoff-design engine behavior, affected by flight conditions.

It can be obtained from full off-design runs or from semi-empirical models.

Flight conditions (altitude and Mach number) are often blended together into asingle parameter: the Dimensionless Freestream Total Temperature θ0

θ0 :=T0a

TSL=

Ta

(1 + (γ−1)

2 M2a

)TSL

(31)

where

0 = total

a = freestream

SL = (Standard day) Sea Level⇒ TSL = 15C

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.126

Constant θ0 contours

note that: θ0 = 1 at sea level static conditions, θ0 can be greater or less than 1,θ0 depends only on Mach number above the tropopause (being T0 constant),the range of θ0 of today’s aircrafts is 0.8 /1.4

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.127

Engine behavior

The engine compressor pressure ratio πc depends upon

the turbine temperature ratio τt

the throttle setting T04

the flight conditions θ0

With the hypothesis of choked turbine (≡ constant temperature ratio τt ), it can

be demonstrated that πc varies only with the ratioT04θ0

.

From the TurboJet power balance [ −(ma + mf )LT = maLC ]:

ηm(1 + f )(h04 − h05) = h03 − h02 (32)

τc − 1 = ηm(1 + f ) (1− τt )1

TSL

(cpt T04)

(cpc θ0); τ :=

h0dn

h0up

; T02 ≈ T0a (33)

⇒ πc = [1 + ηc (τc − 1)]γγ−1 =

(1 + C1

T04

θ0

)γγ−1 (34)

where the constant C1 is:

C1 = ηcηm(1 + f ) (1− τt )1

TSL

cpt

cpc

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.128

Engine behavior (cont’d)

We came up with the relationship between πc and the ratio T04θ0

πc =

(1 + C1

T04

θ0

)γγ−1

it follows that πc increases as the ratio T04θ0

increases.

Example:

with fixed Mach number and throttle setting, πc increases if the aircraftclimbs to higher altitudes (thus θ0 diminishes)

πc diminishes if, for fixed altitude and throttle setting, the aircraft isaccelerated to higher Mach numbers

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.129

Theta break

This image shows the behavior of πc in function of θ0 for fixed values of T04.The horizontal line corresponding to πcmax is due to the intervention of theEngine Control System

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.130

Theta Break (cont’d)

Role of the Engine Control System

Prevent aircraft engines from operating outside their safety margins, such as:

maximum compressor pressure ratio (preventing stall phenomena)

maximum cycle temperature (preventing the turbine to overheat)

These two limits are clearly visible in the previous slide.

The ECS has two logics of operation:when πc is below its maximum allowable value, it limits the maximumallowable temperature reducing πc itself

if πc reaches its maximum, it reduces T04

The value of θ0 at which the control logic switches from limiting πc to limitingT04 is known as:

θ0break

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.131

The importance of Theta Break in the design phase

At any flight condition different from θ0 = θ0break the engine cannot beoperated at its maximum compressor pressure ratio and maximum cycletemperature simultaneously.

If θ0 < θ0break the engine can be operated at πcmax but T04 < T04max(obtaining less specific thrust than nominal)

if θ0 < θ0break the engine can be operated at T04max but πc < πcmax (withgreater fuel consumption than nominal)

The engine designer would therefore have the engine operating as closer aspossible to θ0 = θ0break and this can be done with a wise choice of θ0break

The design with θ0break = 1 implies MAX THRUST AT SEA LEVEL STATIC

A design with θ0break > 1 may be advisable for supersonic aircrafts

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.132

How to design Theta Break?

Recalling from eq. (34) that the control system acts to maintain the ratioTt4θ0

constant in order to keep πc = πcmax = constant, it follows immediately that, inthe case of θ0break ≥ 1:

T04max

θ0break

=

(T04

θ0

)πcmax

=

(T04SL

θ0SL

)πcmax

= T04SL(35)

being θ0SL= 1.

The engine has to be designed to have T04SLgiven by eq. (35), that is the ratio

between the maximum allowable and the desired θ0break .

Defining the Thottle Ratio as the ratio between the maximum allowabletemperature and the sea-level-static temperature:

TR :=T04max

T04SL

= θ0break (36)

it follows that TR is identical to θ0break .

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.133

Throttle Lapse

Example of engine behavior with varying Mach and altitude. It was obtainedwith semi-empirical models of the kind:

α = f [h,M,TR] (TR fixed by design) (37)

Figure: Throttle Lapse of two Low-BPR-TurboFan (one with TR=1.0 and one with TR=1.1)as a function of Mach with varying altitude

Remember that T = αTSL

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.134

Flight Phases

The "master equation" (eq. 30) can be specialized for each flight phase, inorder to obtain the constraints that have to be attained to satisfy the RFP.

The following cases will be analyzed:

Constant altitude/speed cruise

Constant altitude/speed turn

Takeoff Roll

All the other flight phases can be obtained similarly and easily

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.135

Constant altitude/speed cruise

Master equation (eqn. 30):

TSLWTO

= βα

{q

β(WTO/S)

[k1

(nβq (WTO/S)

)2+ k2

(nβq (WTO/S)

)+ CD0

]+ Ps

V

}Known quantities:

Constant altitude and speed⇒ dhdt = 0 dV

dt = 0⇒ PS = 0

Level flight⇒ n = 1 (L = W )

Aerodynamic polar (assigned): k1, k2, CD0 as functions of M, AR, e

Assumptions:

β = 0.7÷ 0.9 (an high enough value to simulate the beginning of thecruise phase, which is the more demanding scenario)

The requirements are given in terms of:

Desired cruise altitude h

Desired cruise Mach number M

Altitude and Mach number appear in:

α(h,M)

q = 12ρ∞(h)V 2

∞ = 12γp∞(h)M2

∞

Aerodynamic polar (CD depends on Mach)

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.136

Constant altitude/speed cruise (cont’d)

Figure: Constant altitude/speed cruise constraint. h = 11000 m, M = 0.85

The minimum of the curve is found at:

[WTO

S

]minT/W

= qβ

√CD0K1

⇒[

TSLWTO

]min

= βα

{2√

CD0 K1 + K2

}which is the condition of minimum thrust and drag (maximum range)

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.137

Constant altitude/speed turn

Master equation (eqn. 30):

TSLWTO

= βα

{q

β(WTO/S)

[k1

(nβq (WTO/S)

)2+ k2

(nβq (WTO/S)

)+ CD0

]+ Ps

V

}Known quantities:

Constant altitude and speed⇒ dhdt = 0 dV

dt = 0⇒ PS = 0

Aerodynamic polar (assigned): k1, k2, CD0 as functions of M, AR, e

Assumptions:

β = 0.7÷ 0.9


Desired cruise altitude h

Desired cruise Mach number M

Desired turn rate n > 1

Remember that during a turn L = nW where n = 1cosφ =

√1 + V 2

g0r (r is theradius of curvature)

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.138

Constant altitude/speed turn (cont’d)

Figure: Constant altitude/speed turn constraint. h = 11000 m, M = 0.85, n = 1.1

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.139

Takeoff Ground RollMaster equation (eqn. 18):

V (T − D)

W=

d(

V 2

2g0+ h)

dt

the takeoff ground roll distance is obtained with the assumption of constantaltitude dh/dt = 0 and TSL >> D:

TSL

WTO=

β

αg0

dVdt

which can be rearranged to (dt = ds/V ):

ds =β

αg0

WTO

TSLVdV

and integrated to yield:

sG =β

αg0

WTO

TSL

V 2TO

2g0

Being:12ρSV 2

STALLCLmax = βWTO

where VTO = kTOVSTALL and kTO ≈ 1.2, the equation can be recast to give:

sG =β2

α

k2TO

(TSL/WTO) ρ g0CLmax

(WTO

S

)

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.140

Takeoff Ground Roll (cont’d)

The ground roll distance equation:

sG =β2

α

k2TO

(TSL/WTO) ρ g0CLmax

(WTO

S

)tells that the ground roll distance is inversely proportional to:

density ρ, which is function of altitude and temperature. High elevationairports and hot days deteriorate take-off performance.

aerodynamic performance CLmax , whose value is increased by the useof high lift devices (slats, flaps, ...)

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.141

Solving the previous equation for TSL/WTO , the take-off constraint as afunction of WTO/S can be obtained:

TSL

WTO=β2

α

k2TO

sG ρ g0CLmax

(WTO

S

)(38)

Known quantities:

CLmax (≈ 1.5÷ 2.7 depending on installed H-L-devices)

Assumptions:

β = 1 (MTOW)


Desired field elevation and temperature⇒ ρ[h,T ]

Desired Takeoff speed (VTO = kTO VSTALL, es: kTO = 1.2)

Field length available sG

Note that airport elevation and temperature affect also the engine performancethrough α(h,M0)

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.142

Takeoff Roll (cont’d)

The relationship is linear

Figure: Takeoff Roll. h = 1500 m, CLmax = 2.4, sG = 2000 m, HOT (ISA +30) vs COLD(ISA +0) day

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.143

Constraint diagram

Cases analyzed: constant altitude/speed cruise, constant altitude/speed turn,takeoff roll, horizontal acceleration, service ceiling

Figure: All constraints

The solution space is the white region over the constraint curves. We canchoose:

TSL

WTOand

WTO

S

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.144

Aim of the mission analysis

Constraint analysis⇒ TSLWTO

and WTOS

Next step⇒ establish the size of the aircraft via the estimation of themaximum takeoff weight WTO (o MTOW)

With the MTOW in hand, the Sea level Max Thrust TSL and the wing area S areimmediately obtained:

TSL

WTO·WTO = TSL(

WTO

S

)−1·WTO = S

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.145

Who is WTO?

WTO is the sum of 3 main contributions:empty weight WE

aircraft structuresequipments (engines, avionics, seats, etc.)

payload weight WP

fuel weight WF

WTO = WE + WP + WF (39)

The sum of Empty weight and Payload weight is the Maximum Zero FuelWeight (MZFW)

it follows that:WTO =

WP

1− WFWTO− WE

WTO

(40)

The fuel weight WF represents the fuel gradually consumed during themission: the aircraft weight decreases at exactly the same rate at the which thefuel is consumed:

dWdt

= −dWF

dt= −TSFC · T (41)

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.146

Cruise weight ratio

We need WF/WTO , given desired range and payload (from RFP).

For the cruise flight phase, the required thrust is not known, being it throttleddown so that:

T = D(h,M,AOA)

Equation (41) becomes:

dWW

= −TSFCDW

dt = −TSFCDW

dsV

(42)

The integration requires the knowledge of the behavior of the term −TSFC DW

during the mission.

It is often found that this term remains relatively unaltered over the flight leg,and the integral can be approximated as:

Wf

Wi= exp

(−TSFC

(DW

)∆t)

(43)

where ∆t is the cruise flight time.

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.147

TSFC behavior

We now seek the behavior of TSFC with varying flight conditions.

TSFC is a complex function of the combination of instantaneous altitude,speed and throttle setting

A satisfactory approximation for this design stage is the following:

TSFC ≈ (C1 + C2M) (44)

where C1 and C2 are constants that are known in advance for each type ofengine cycle.

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.148

Best Subsonic Cruise

Subsonic cruise is usually the most important portion of any missionbecause it uses the largest amount of onboard fuel.

Eqn. (42) can be simplified to yield:

dWW

= −C1/M + C2

astd

(CD

CL

)ds (45)

where astd is the speed of sound for ISA conditions.Best Cruise condition⇒ minimum fuel consumption⇒

min[

(C1/M + C2)

(CD

CL

)]= (C1/M? + C2)

(C?DC?L

)

The best cruise is identified by the values M?, C?L , C?D

Next task: find their expressions

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.149

Best Subsonic Cruise Mach number

Below the critical drag rise Mach number MCRIT (approximately 0.8), neitherCL nor CD depend on M.

In this range:CD

CL=

CD0 + K1C2L + K2CL

CL(46)

whose minimum can be found differentiating with respect to CL:

(CD

CL

)∗=√

4K1(CD0

)+ K2 at C∗L =

√CD0

K1(47)

which is the maximum efficiency CL, as expected.

Next, because the lowest achievable value of(

CDCL

)is constant below MCRIT , it

follows that:

(C1/M + C2)(

CDCL

)decreases as M increases

Further increases past MCRIT cause(

CDCL

)to increase again

⇒ M? = MCRIT

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.150

Best Subsonic Cruise Weight Fraction

The final equation is:

dWW

= −(

C1

MCRIT+ C2

)(√4(CD0 )K1 + K2

) dsastd

(48)

whose integral is exact and yields:

ΠCRZ =Wf

Wi= exp

[−(

C1

MCRIT+ C2

)(√4(CD0 )K1 + K2

)] ∆sastd

(49)

Knowing:

the desired range ∆s

the values for the TSFC constants C1 and C2

the aerodynamic polar

we can estimate the cruise fuel depletion.

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.151

Best Subsonic Cruise AltitudeThe altitude may not be arbitrarily chosen

It is the altitude that allows the condition of self-sustained flight with thosevalues of M?, C?L and C?D . Being:

L = βWTO ⇒12γM2pSC?L = βWTO

Recalling that C?L =√

CD0/K1, it follows that:

δ :=p

pstd=

1pstd

2βγM2

CRIT

1√(CD0 )/K1

(WTO

S

)(50)

Remember that β decreases during cruise, so:δ (⇒ altitude) must gradually increasebeing M fixed to M?, the speed must gradually decrease until thetropopause is reached

Usually steps of 2000 ft are employed, due to air traffic control concerns.

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.152


Recalling eqn. 39WTO = WE + WP + WF (51)

it follows that:WTO =

WP

1− WFWTO− WE

WTO

(52)

WP is the desired payload weightWFWTO

is the product of the weight ratios of all the phases of flight (weexamined best cruise only)WEWTO

can be estimated with empirical models as a function of WTO

N.B. The calculations of WTO requires an iterative procedure because of thedependence of WE

WTOon WTO

Aircraft Mission



Mathematical model

Aerodynamic Polar

Throttle Lapse

Flight phases


Aircraft weights

Cruise weight ratio

TSFC behavior

BCM/BCA


Conclusions

8.153

What we have done so far and what’s next

Done:

RFP⇒ Constraints on TSL/WTO and WTO/S

First attempt choice of TSL/WTO and WTO/S

Mission Analysis⇒WTO ⇒ TSL

Next: How to achieve this TSL?

We have performance to meet (TSFC)

We have design limitations (T4max , βCmax , etc.)

We can play with design parameters (BPR, βF , βC , etc.)

⇒ Parametric Cycle Analysis and Performance estimation

aircraft mission lecture 8 aircraft...

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